We identify the representation space of an arbitrarily oriented type A quiver with an open subscheme of a Schubert variety, up to a smooth factor,
with all orbit closures being given by Schubert conditions. Consequently, the space has a Frobenius splitting which
compatibly splits the orbit closures, and we can realize the orbit closure poset as a subposet of the symmetric group under Bruhat order.
This also gives an alternative proof of Bobiński and Zwara's result that these orbit closures
are normal, Cohen-Macaulay, and have rational singularities.

In this paper we seek invariant-theoretic characterizations of (Schur-)representation finite algebras. To this end, we introduce two classes of finite-dimensional algebras: those with the dense-orbit property and those
with the multiplicity-free property. We show first that when a connected algebra A admits a pre-projective component, each of these properties is equivalent to A being representation-finite. Next, we give an example of a
representation-infinite algebra with the dense-orbit property. We also show that the string algebras with the dense orbit-property are precisely the representation-finite ones. Finally, we show that a tame algebra has the
multiplicity-free property if and only if it is Schur-representation-finite.

We give a formula for counting tree modules for the quiver S_g with g loops and one vertex in terms of tree modules on its universal cover. This formula, along with work of Helleloid and Rodriguez-Villegas, is used to show that the number of d-dimensional tree modules for S_g is polynomial in g with the same degree and leading coefficient as the counting polynomial A_{S_g}(d, q) for absolutely indecomposables over F_q, evaluated at q=1.

We give a general procedure for taking a collection of directed graphs and maps between them to construct idempotents in representation rings, then show how to combinatorially orthogonalize them using a categorical version of Möbius inversion. This is applied to a category of directed graphs which gives orthogonal idempotents associated to all representations of a fixed quiver which are projective or injective on some subquiver. For more detail, see notes from 2010 Auslander conference below.

"Rank Loci in Representation Spaces of Quivers", preprint available at arXiv:1004.1981.

An infinite hierarchy of inequalities that hold for the rank function of a subspace arrangement is given. Previously, only one non-elementary inequality for subspace arrangements was known. It gives new criteria for realizability (linear representability) of matroids. (See also the work of Dougherty, Freiling, and Zeger: arxiv:0910.0284.)

This essentially a concatenation of the two papers below, improved with the benefit of hindsight, more readers, and lack of space limitations (more examples, more background, and remarks on generalizations).

A functor is constructed that generalizes the rank of a linear map to the setting of an arbitrary diagram of vector spaces and linear maps (a quiver representation). Using maps of directed graphs, we get more, similar functors. These can be used to construct numerical invariants of a quiver representation which include, as the simplest cases, its dimension vector and the ranks of all maps appearing in the representation.

Notes and slides from various talks

Slides from ICRA 2012 in Bielefeld on modules varieties with dense orbits in every component and generic representation theory of algebras.

Notes from a short talk on the tensor products of quiver representations and forbidden minors for MFT (AMS Syracuse, Fall 2010).

Some fractals that I made as part of undergrad summer research with Estela Gavosto at the University of Kansas. These are (complex) one-dimensional slices of a (complex) two-dimensional parameter space arising from the Hénon map. The classical Mandelbrot set is, for example, one of the one-dimensional slices of this set, thus the similarity to some of these slices.